Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems

TL;DR

This paper introduces a new family of Bell inequalities for high-dimensional quantum systems that achieve maximal violation with maximally entangled states and mutually unbiased bases, enabling self-testing of two-qutrit systems.

Contribution

The authors propose a novel class of Bell inequalities with analytically computable maximal violations that generalize CHSH and facilitate self-testing of two-qutrit maximally entangled states.

Findings

Maximal violation achieved by maximally entangled states and MUBs.

In the three-outcome case, Bell inequalities enable self-testing of two-qutrit states.

For higher outcomes, maximal violation does not allow standard self-testing, leading to a new weak self-testing concept.

Abstract

Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve $d$ measurements settings, each having $d$ outcomes for an arbitrary prime number $d\geq 3$. We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.